//************************************************************************************
// BigInteger Class Version 1.03
//
// Copyright (c) 2002 Chew Keong TAN
// All rights reserved.
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the
// "Software"), to deal in the Software without restriction, including
// without limitation the rights to use, copy, modify, merge, publish,
// distribute, and/or sell copies of the Software, and to permit persons
// to whom the Software is furnished to do so, provided that the above
// copyright notice(s) and this permission notice appear in all copies of
// the Software and that both the above copyright notice(s) and this
// permission notice appear in supporting documentation.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
// OF THIRD PARTY RIGHTS. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR
// HOLDERS INCLUDED IN THIS NOTICE BE LIABLE FOR ANY CLAIM, OR ANY SPECIAL
// INDIRECT OR CONSEQUENTIAL DAMAGES, OR ANY DAMAGES WHATSOEVER RESULTING
// FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT,
// NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION
// WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
//
//
// Disclaimer
// ----------
// Although reasonable care has been taken to ensure the correctness of this
// implementation, this code should never be used in any application without
// proper verification and testing.  I disclaim all liability and responsibility
// to any person or entity with respect to any loss or damage caused, or alleged
// to be caused, directly or indirectly, by the use of this BigInteger class.
//
// Comments, bugs and suggestions to
// (http://www.codeproject.com/csharp/biginteger.asp)
//
//
// Overloaded Operators +, -, *, /, %, >>, <<, ==, !=, >, <, >=, <=, &, |, ^, ++, --, ~
//
// Features
// --------
// 1) Arithmetic operations involving large signed integers (2's complement).
// 2) Primality test using Fermat little theorm, Rabin Miller's method,
//    Solovay Strassen's method and Lucas strong pseudoprime.
// 3) Modulo exponential with Barrett's reduction.
// 4) Inverse modulo.
// 5) Pseudo prime generation.
// 6) Co-prime generation.
//
//
// Known Problem
// -------------
// This pseudoprime passes my implementation of
// primality test but failed in JDK's isProbablePrime test.
//
//       byte[] pseudoPrime1 = { (byte)0x00,
//             (byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
//             (byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
//             (byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
//             (byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
//             (byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
//             (byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
//             (byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
//             (byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
//             (byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
//             (byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
//             (byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
//       };
//
//
// Change Log
// ----------
// 1) September 23, 2002 (Version 1.03)
//    - Fixed operator- to give correct data length.
//    - Added Lucas sequence generation.
//    - Added Strong Lucas Primality test.
//    - Added integer square root method.
//    - Added setBit/unsetBit methods.
//    - New isProbablePrime() method which do not require the
//      confident parameter.
//
// 2) August 29, 2002 (Version 1.02)
//    - Fixed bug in the exponentiation of negative numbers.
//    - Faster modular exponentiation using Barrett reduction.
//    - Added getBytes() method.
//    - Fixed bug in ToHexString method.
//    - Added overloading of ^ operator.
//    - Faster computation of Jacobi symbol.
//
// 3) August 19, 2002 (Version 1.01)
//    - Big integer is stored and manipulated as unsigned integers (4 bytes) instead of
//      individual bytes this gives significant performance improvement.
//    - Updated Fermat's Little Theorem test to use a^(p-1) mod p = 1
//    - Added isProbablePrime method.
//    - Updated documentation.
//
// 4) August 9, 2002 (Version 1.0)
//    - Initial Release.
//
//
// References
// [1] D. E. Knuth, "Seminumerical Algorithms", The Art of Computer Programming Vol. 2,
//     3rd Edition, Addison-Wesley, 1998.
//
// [2] K. H. Rosen, "Elementary Number Theory and Its Applications", 3rd Ed,
//     Addison-Wesley, 1993.
//
// [3] B. Schneier, "Applied Cryptography", 2nd Ed, John Wiley & Sons, 1996.
//
// [4] A. Menezes, P. van Oorschot, and S. Vanstone, "Handbook of Applied Cryptography",
//     CRC Press, 1996, www.cacr.math.uwaterloo.ca/hac
//
// [5] A. Bosselaers, R. Govaerts, and J. Vandewalle, "Comparison of Three Modular
//     Reduction Functions," Proc. CRYPTO'93, pp.175-186.
//
// [6] R. Baillie and S. S. Wagstaff Jr, "Lucas Pseudoprimes", Mathematics of Computation,
//     Vol. 35, No. 152, Oct 1980, pp. 1391-1417.
//
// [7] H. C. Williams, "Édouard Lucas and Primality Testing", Canadian Mathematical
//     Society Series of Monographs and Advance Texts, vol. 22, John Wiley & Sons, New York,
//     NY, 1998.
//
// [8] P. Ribenboim, "The new book of prime number records", 3rd edition, Springer-Verlag,
//     New York, NY, 1995.
//
// [9] M. Joye and J.-J. Quisquater, "Efficient computation of full Lucas sequences",
//     Electronics Letters, 32(6), 1996, pp 537-538.
//
//************************************************************************************

using System;


public class BigInteger
{
        // maximum length of the BigInteger in uint (4 bytes)
        // change this to suit the required level of precision.

        private const int maxLength = 70;

        // primes smaller than 2000 to test the generated prime number

        public static readonly int[] primesBelow2000 = {
        2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
        101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
	211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
	307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
	401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
	503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,
	601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
	701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,
	809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
	907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
	1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097,
	1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193,
	1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,
	1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399,
	1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499,
	1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
	1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699,
	1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789,
	1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
	1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999 };


        private uint[] data = null;             // stores bytes from the Big Integer
        public int dataLength;                 // number of actual chars used


        //***********************************************************************
        // Constructor (Default value for BigInteger is 0
        //***********************************************************************

        public BigInteger()
        {
                data = new uint[maxLength];
                dataLength = 1;
        }


        //***********************************************************************
        // Constructor (Default value provided by long)
        //***********************************************************************

        public BigInteger(long value)
        {
                data = new uint[maxLength];
                long tempVal = value;

                // copy bytes from long to BigInteger without any assumption of
                // the length of the long datatype

                dataLength = 0;
                while(value != 0 && dataLength < maxLength)
                {
                        data[dataLength] = (uint)(value & 0xFFFFFFFF);
                        value >>= 32;
                        dataLength++;
                }

                if(tempVal > 0)         // overflow check for +ve value
                {
                        if(value != 0 || (data[maxLength-1] & 0x80000000) != 0)
                                throw(new ArithmeticException("Positive overflow in constructor."));
                }
                else if(tempVal < 0)    // underflow check for -ve value
                {
                        if(value != -1 || (data[dataLength-1] & 0x80000000) == 0)
                                throw(new ArithmeticException("Negative underflow in constructor."));
                }

                if(dataLength == 0)
                        dataLength = 1;
        }


        //***********************************************************************
        // Constructor (Default value provided by ulong)
        //***********************************************************************

        public BigInteger(ulong value)
        {
                data = new uint[maxLength];

                // copy bytes from ulong to BigInteger without any assumption of
                // the length of the ulong datatype

                dataLength = 0;
                while(value != 0 && dataLength < maxLength)
                {
                        data[dataLength] = (uint)(value & 0xFFFFFFFF);
                        value >>= 32;
                        dataLength++;
                }

                if(value != 0 || (data[maxLength-1] & 0x80000000) != 0)
                        throw(new ArithmeticException("Positive overflow in constructor."));

                if(dataLength == 0)
                        dataLength = 1;
        }



        //***********************************************************************
        // Constructor (Default value provided by BigInteger)
        //***********************************************************************

        public BigInteger(BigInteger bi)
        {
                data = new uint[maxLength];

                dataLength = bi.dataLength;

                for(int i = 0; i < dataLength; i++)
                        data[i] = bi.data[i];
        }


        //***********************************************************************
        // Constructor (Default value provided by a string of digits of the
        //              specified base)
        //
        // Example (base 10)
        // -----------------
        // To initialize "a" with the default value of 1234 in base 10
        //      BigInteger a = new BigInteger("1234", 10)
        //
        // To initialize "a" with the default value of -1234
        //      BigInteger a = new BigInteger("-1234", 10)
        //
        // Example (base 16)
        // -----------------
        // To initialize "a" with the default value of 0x1D4F in base 16
        //      BigInteger a = new BigInteger("1D4F", 16)
        //
        // To initialize "a" with the default value of -0x1D4F
        //      BigInteger a = new BigInteger("-1D4F", 16)
        //
        // Note that string values are specified in the <sign><magnitude>
        // format.
        //
        //***********************************************************************

        public BigInteger(string value, int radix)
        {
                BigInteger multiplier = new BigInteger(1);
                BigInteger result = new BigInteger();
                value = (value.ToUpper()).Trim();
                int limit = 0;

                if(value[0] == '-')
                        limit = 1;

                for(int i = value.Length - 1; i >= limit ; i--)
                {
                        int posVal = (int)value[i];

                        if(posVal >= '0' && posVal <= '9')
                                posVal -= '0';
                        else if(posVal >= 'A' && posVal <= 'Z')
                                posVal = (posVal - 'A') + 10;
                        else
                                posVal = 9999999;       // arbitrary large


                        if(posVal >= radix)
                                throw(new ArithmeticException("Invalid string in constructor."));
                        else
                        {
                                if(value[0] == '-')
                                        posVal = -posVal;

                                result = result + (multiplier * posVal);

                                if((i - 1) >= limit)
                                        multiplier = multiplier * radix;
                        }
                }

                if(value[0] == '-')     // negative values
                {
                        if((result.data[maxLength-1] & 0x80000000) == 0)
                                throw(new ArithmeticException("Negative underflow in constructor."));
                }
                else    // positive values
                {
                        if((result.data[maxLength-1] & 0x80000000) != 0)
                                throw(new ArithmeticException("Positive overflow in constructor."));
                }

                data = new uint[maxLength];
                for(int i = 0; i < result.dataLength; i++)
                        data[i] = result.data[i];

                dataLength = result.dataLength;
        }


        //***********************************************************************
        // Constructor (Default value provided by an array of bytes)
        //
        // The lowest index of the input byte array (i.e [0]) should contain the
        // most significant byte of the number, and the highest index should
        // contain the least significant byte.
        //
        // E.g.
        // To initialize "a" with the default value of 0x1D4F in base 16
        //      byte[] temp = { 0x1D, 0x4F };
        //      BigInteger a = new BigInteger(temp)
        //
        // Note that this method of initialization does not allow the
        // sign to be specified.
        //
        //***********************************************************************

        public BigInteger(byte[] inData)
        {
                dataLength = inData.Length >> 2;

                int leftOver = inData.Length & 0x3;
                if(leftOver != 0)         // length not multiples of 4
                        dataLength++;


                if(dataLength > maxLength)
                      throw(new ArithmeticException("Byte overflow in constructor."));

                data = new uint[maxLength];

                for(int i = inData.Length - 1, j = 0; i >= 3; i -= 4, j++)
                {
                        data[j] = (uint)((inData[i-3] << 24) + (inData[i-2] << 16) +
                                         (inData[i-1] <<  8) + inData[i]);
                }

                if(leftOver == 1)
                        data[dataLength-1] = (uint)inData[0];
                else if(leftOver == 2)
                        data[dataLength-1] = (uint)((inData[0] << 8) + inData[1]);
                else if(leftOver == 3)
                        data[dataLength-1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);


                while(dataLength > 1 && data[dataLength-1] == 0)
                        dataLength--;

                //Console.WriteLine("Len = " + dataLength);
        }


        //***********************************************************************
        // Constructor (Default value provided by an array of bytes of the
        // specified length.)
        //***********************************************************************

        public BigInteger(byte[] inData, int inLen)
        {
                dataLength = inLen >> 2;

                int leftOver = inLen & 0x3;
                if(leftOver != 0)         // length not multiples of 4
                        dataLength++;

                if(dataLength > maxLength || inLen > inData.Length)
                      throw(new ArithmeticException("Byte overflow in constructor."));


                data = new uint[maxLength];

                for(int i = inLen - 1, j = 0; i >= 3; i -= 4, j++)
                {
                        data[j] = (uint)((inData[i-3] << 24) + (inData[i-2] << 16) +
                                         (inData[i-1] <<  8) + inData[i]);
                }

                if(leftOver == 1)
                        data[dataLength-1] = (uint)inData[0];
                else if(leftOver == 2)
                        data[dataLength-1] = (uint)((inData[0] << 8) + inData[1]);
                else if(leftOver == 3)
                        data[dataLength-1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);


                if(dataLength == 0)
                        dataLength = 1;

                while(dataLength > 1 && data[dataLength-1] == 0)
                        dataLength--;

                //Console.WriteLine("Len = " + dataLength);
        }


        //***********************************************************************
        // Constructor (Default value provided by an array of unsigned integers)
        //*********************************************************************

        public BigInteger(uint[] inData)
        {
                dataLength = inData.Length;

                if(dataLength > maxLength)
                      throw(new ArithmeticException("Byte overflow in constructor."));

                data = new uint[maxLength];

                for(int i = dataLength - 1, j = 0; i >= 0; i--, j++)
                        data[j] = inData[i];

                while(dataLength > 1 && data[dataLength-1] == 0)
                        dataLength--;

                //Console.WriteLine("Len = " + dataLength);
        }


        //***********************************************************************
        // Overloading of the typecast operator.
        // For BigInteger bi = 10;
        //***********************************************************************

        public static implicit operator BigInteger(long value)
        {
                return (new BigInteger(value));
        }

        public static implicit operator BigInteger(ulong value)
        {
                return (new BigInteger(value));
        }

        public static implicit operator BigInteger(int value)
        {
                return (new BigInteger((long)value));
        }

        public static implicit operator BigInteger(uint value)
        {
                return (new BigInteger((ulong)value));
        }


        //***********************************************************************
        // Overloading of addition operator
        //***********************************************************************

        public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
        {
                BigInteger result = new BigInteger();

                result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

                long carry = 0;
                for(int i = 0; i < result.dataLength; i++)
                {
                        long sum = (long)bi1.data[i] + (long)bi2.data[i] + carry;
                        carry  = sum >> 32;
                        result.data[i] = (uint)(sum & 0xFFFFFFFF);
                }

                if(carry != 0 && result.dataLength < maxLength)
                {
                        result.data[result.dataLength] = (uint)(carry);
                        result.dataLength++;
                }

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;


                // overflow check
                int lastPos = maxLength - 1;
                if((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
                   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
                {
                        throw (new ArithmeticException());
                }

                return result;
        }


        //***********************************************************************
        // Overloading of the unary ++ operator
        //***********************************************************************

        public static BigInteger operator ++(BigInteger bi1)
        {
                BigInteger result = new BigInteger(bi1);

                long val, carry = 1;
                int index = 0;

                while(carry != 0 && index < maxLength)
                {
                        val = (long)(result.data[index]);
                        val++;

                        result.data[index] = (uint)(val & 0xFFFFFFFF);
                        carry = val >> 32;

                        index++;
                }

                if(index > result.dataLength)
                        result.dataLength = index;
                else
                {
                        while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                                result.dataLength--;
                }

                // overflow check
                int lastPos = maxLength - 1;

                // overflow if initial value was +ve but ++ caused a sign
                // change to negative.

                if((bi1.data[lastPos] & 0x80000000) == 0 &&
                   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
                {
                        throw (new ArithmeticException("Overflow in ++."));
                }
                return result;
        }


        //***********************************************************************
        // Overloading of subtraction operator
        //***********************************************************************

        public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
        {
                BigInteger result = new BigInteger();

                result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

                long carryIn = 0;
                for(int i = 0; i < result.dataLength; i++)
                {
                        long diff;

                        diff = (long)bi1.data[i] - (long)bi2.data[i] - carryIn;
                        result.data[i] = (uint)(diff & 0xFFFFFFFF);

                        if(diff < 0)
                                carryIn = 1;
                        else
                                carryIn = 0;
                }

                // roll over to negative
                if(carryIn != 0)
                {
                        for(int i = result.dataLength; i < maxLength; i++)
                                result.data[i] = 0xFFFFFFFF;
                        result.dataLength = maxLength;
                }

                // fixed in v1.03 to give correct datalength for a - (-b)
                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                // overflow check

                int lastPos = maxLength - 1;
                if((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) &&
                   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
                {
                        throw (new ArithmeticException());
                }

                return result;
        }


        //***********************************************************************
        // Overloading of the unary -- operator
        //***********************************************************************

        public static BigInteger operator --(BigInteger bi1)
        {
                BigInteger result = new BigInteger(bi1);

                long val;
                bool carryIn = true;
                int index = 0;

                while(carryIn && index < maxLength)
                {
                        val = (long)(result.data[index]);
                        val--;

                        result.data[index] = (uint)(val & 0xFFFFFFFF);

                        if(val >= 0)
                                carryIn = false;

                        index++;
                }

                if(index > result.dataLength)
                        result.dataLength = index;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                // overflow check
                int lastPos = maxLength - 1;

                // overflow if initial value was -ve but -- caused a sign
                // change to positive.

                if((bi1.data[lastPos] & 0x80000000) != 0 &&
                   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
                {
                        throw (new ArithmeticException("Underflow in --."));
                }

                return result;
        }


        //***********************************************************************
        // Overloading of multiplication operator
        //***********************************************************************

        public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
        {
                int lastPos = maxLength-1;
                bool bi1Neg = false, bi2Neg = false;

                // take the absolute value of the inputs
                try
                {
                        if((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
                        {
                                bi1Neg = true; bi1 = -bi1;
                        }
                        if((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
                        {
                                bi2Neg = true; bi2 = -bi2;
                        }
                }
                catch(Exception) {}

                BigInteger result = new BigInteger();

                // multiply the absolute values
                try
                {
                        for(int i = 0; i < bi1.dataLength; i++)
                        {
                                if(bi1.data[i] == 0)    continue;

                                ulong mcarry = 0;
                                for(int j = 0, k = i; j < bi2.dataLength; j++, k++)
                                {
                                        // k = i + j
                                        ulong val = ((ulong)bi1.data[i] * (ulong)bi2.data[j]) +
                                                     (ulong)result.data[k] + mcarry;

                                        result.data[k] = (uint)(val & 0xFFFFFFFF);
			                mcarry = (val >> 32);
                                }

                                if(mcarry != 0)
                                        result.data[i+bi2.dataLength] = (uint)mcarry;
                        }
                }
                catch(Exception)
                {
                        throw(new ArithmeticException("Multiplication overflow."));
                }


                result.dataLength = bi1.dataLength + bi2.dataLength;
                if(result.dataLength > maxLength)
                        result.dataLength = maxLength;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                // overflow check (result is -ve)
                if((result.data[lastPos] & 0x80000000) != 0)
                {
                        if(bi1Neg != bi2Neg && result.data[lastPos] == 0x80000000)    // different sign
                        {
                                // handle the special case where multiplication produces
                                // a max negative number in 2's complement.

                                if(result.dataLength == 1)
                                        return result;
                                else
                                {
                                        bool isMaxNeg = true;
                                        for(int i = 0; i < result.dataLength - 1 && isMaxNeg; i++)
                                        {
                                                if(result.data[i] != 0)
                                                        isMaxNeg = false;
                                        }

                                        if(isMaxNeg)
                                                return result;
                                }
                        }

                        throw(new ArithmeticException("Multiplication overflow."));
                }

                // if input has different signs, then result is -ve
                if(bi1Neg != bi2Neg)
                        return -result;

                return result;
        }



        //***********************************************************************
        // Overloading of unary << operators
        //***********************************************************************

        public static BigInteger operator <<(BigInteger bi1, int shiftVal)
        {
                BigInteger result = new BigInteger(bi1);
                result.dataLength = shiftLeft(result.data, shiftVal);

                return result;
        }


        // least significant bits at lower part of buffer

        private static int shiftLeft(uint[] buffer, int shiftVal)
        {
                int shiftAmount = 32;
                int bufLen = buffer.Length;

                while(bufLen > 1 && buffer[bufLen-1] == 0)
                        bufLen--;

                for(int count = shiftVal; count > 0;)
                {
                        if(count < shiftAmount)
                                shiftAmount = count;

                        //Console.WriteLine("shiftAmount = {0}", shiftAmount);

                        ulong carry = 0;
                        for(int i = 0; i < bufLen; i++)
                        {
                                ulong val = ((ulong)buffer[i]) << shiftAmount;
                                val |= carry;

                                buffer[i] = (uint)(val & 0xFFFFFFFF);
                                carry = val >> 32;
                        }

                        if(carry != 0)
                        {
                                if(bufLen + 1 <= buffer.Length)
                                {
                                        buffer[bufLen] = (uint)carry;
                                        bufLen++;
                                }
                        }
                        count -= shiftAmount;
                }
                return bufLen;
        }


        //***********************************************************************
        // Overloading of unary >> operators
        //***********************************************************************

        public static BigInteger operator >>(BigInteger bi1, int shiftVal)
        {
                BigInteger result = new BigInteger(bi1);
                result.dataLength = shiftRight(result.data, shiftVal);


                if((bi1.data[maxLength-1] & 0x80000000) != 0) // negative
                {
                        for(int i = maxLength - 1; i >= result.dataLength; i--)
                                result.data[i] = 0xFFFFFFFF;

                        uint mask = 0x80000000;
                        for(int i = 0; i < 32; i++)
                        {
                                if((result.data[result.dataLength-1] & mask) != 0)
                                        break;

                                result.data[result.dataLength-1] |= mask;
                                mask >>= 1;
                        }
                        result.dataLength = maxLength;
                }

                return result;
        }


        private static int shiftRight(uint[] buffer, int shiftVal)
        {
                int shiftAmount = 32;
                int invShift = 0;
                int bufLen = buffer.Length;

                while(bufLen > 1 && buffer[bufLen-1] == 0)
                        bufLen--;

                //Console.WriteLine("bufLen = " + bufLen + " buffer.Length = " + buffer.Length);

                for(int count = shiftVal; count > 0;)
                {
                        if(count < shiftAmount)
                        {
                                shiftAmount = count;
                                invShift = 32 - shiftAmount;
                        }

                        //Console.WriteLine("shiftAmount = {0}", shiftAmount);

                        ulong carry = 0;
                        for(int i = bufLen - 1; i >= 0; i--)
                        {
                                ulong val = ((ulong)buffer[i]) >> shiftAmount;
                                val |= carry;

                                carry = ((ulong)buffer[i]) << invShift;
                                buffer[i] = (uint)(val);
                        }

                        count -= shiftAmount;
                }

                while(bufLen > 1 && buffer[bufLen-1] == 0)
                        bufLen--;

                return bufLen;
        }


        //***********************************************************************
        // Overloading of the NOT operator (1's complement)
        //***********************************************************************

        public static BigInteger operator ~(BigInteger bi1)
        {
                BigInteger result = new BigInteger(bi1);

                for(int i = 0; i < maxLength; i++)
                        result.data[i] = (uint)(~(bi1.data[i]));

                result.dataLength = maxLength;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                return result;
        }


        //***********************************************************************
        // Overloading of the NEGATE operator (2's complement)
        //***********************************************************************

        public static BigInteger operator -(BigInteger bi1)
        {
                // handle neg of zero separately since it'll cause an overflow
                // if we proceed.

                if(bi1.dataLength == 1 && bi1.data[0] == 0)
                        return (new BigInteger());

                BigInteger result = new BigInteger(bi1);

                // 1's complement
                for(int i = 0; i < maxLength; i++)
                        result.data[i] = (uint)(~(bi1.data[i]));

                // add one to result of 1's complement
                long val, carry = 1;
                int index = 0;

                while(carry != 0 && index < maxLength)
                {
                        val = (long)(result.data[index]);
                        val++;

                        result.data[index] = (uint)(val & 0xFFFFFFFF);
                        carry = val >> 32;

                        index++;
                }

                if((bi1.data[maxLength-1] & 0x80000000) == (result.data[maxLength-1] & 0x80000000))
                        throw (new ArithmeticException("Overflow in negation.\n"));

                result.dataLength = maxLength;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;
                return result;
        }


        //***********************************************************************
        // Overloading of equality operator
        //***********************************************************************

        public static bool operator ==(BigInteger bi1, BigInteger bi2)
        {
                return bi1.Equals(bi2);
        }


        public static bool operator !=(BigInteger bi1, BigInteger bi2)
        {
                return !(bi1.Equals(bi2));
        }


        public override bool Equals(object o)
        {
                BigInteger bi = (BigInteger)o;

                if(this.dataLength != bi.dataLength)
                        return false;

                for(int i = 0; i < this.dataLength; i++)
                {
                        if(this.data[i] != bi.data[i])
                                return false;
                }
                return true;
        }


        public override int GetHashCode()
        {
                return this.ToString().GetHashCode();
        }


        //***********************************************************************
        // Overloading of inequality operator
        //***********************************************************************

        public static bool operator >(BigInteger bi1, BigInteger bi2)
        {
        	int pos = maxLength - 1;

        	// bi1 is negative, bi2 is positive
        	if((bi1.data[pos] & 0x80000000) != 0 && (bi2.data[pos] & 0x80000000) == 0)
        	        return false;

                // bi1 is positive, bi2 is negative
        	else if((bi1.data[pos] & 0x80000000) == 0 && (bi2.data[pos] & 0x80000000) != 0)
        	        return true;

                // same sign
        	int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
	        for(pos = len - 1; pos >= 0 && bi1.data[pos] == bi2.data[pos]; pos--);

	        if(pos >= 0)
	        {
		        if(bi1.data[pos] > bi2.data[pos])
			        return true;
		        return false;
	        }
	        return false;
        }


        public static bool operator <(BigInteger bi1, BigInteger bi2)
        {
        	int pos = maxLength - 1;

        	// bi1 is negative, bi2 is positive
        	if((bi1.data[pos] & 0x80000000) != 0 && (bi2.data[pos] & 0x80000000) == 0)
        	        return true;

                // bi1 is positive, bi2 is negative
        	else if((bi1.data[pos] & 0x80000000) == 0 && (bi2.data[pos] & 0x80000000) != 0)
        	        return false;

        	// same sign
        	int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
	        for(pos = len - 1; pos >= 0 && bi1.data[pos] == bi2.data[pos]; pos--);

	        if(pos >= 0)
	        {
		        if(bi1.data[pos] < bi2.data[pos])
			        return true;
		        return false;
	        }
	        return false;
        }


        public static bool operator >=(BigInteger bi1, BigInteger bi2)
        {
                return (bi1 == bi2 || bi1 > bi2);
        }


        public static bool operator <=(BigInteger bi1, BigInteger bi2)
        {
                return (bi1 == bi2 || bi1 < bi2);
        }


        //***********************************************************************
        // Private function that supports the division of two numbers with
        // a divisor that has more than 1 digit.
        //
        // Algorithm taken from [1]
        //***********************************************************************

        private static void multiByteDivide(BigInteger bi1, BigInteger bi2,
                                            BigInteger outQuotient, BigInteger outRemainder)
        {
                uint[] result = new uint[maxLength];

                int remainderLen = bi1.dataLength + 1;
                uint[] remainder = new uint[remainderLen];

                uint mask = 0x80000000;
                uint val = bi2.data[bi2.dataLength - 1];
                int shift = 0, resultPos = 0;

                while(mask != 0 && (val & mask) == 0)
                {
                        shift++; mask >>= 1;
                }

                //Console.WriteLine("shift = {0}", shift);
                //Console.WriteLine("Before bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);

                for(int i = 0; i < bi1.dataLength; i++)
                        remainder[i] = bi1.data[i];
                shiftLeft(remainder, shift);
                bi2 = bi2 << shift;

                /*
                Console.WriteLine("bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);
                Console.WriteLine("dividend = " + bi1 + "\ndivisor = " + bi2);
                for(int q = remainderLen - 1; q >= 0; q--)
                        Console.Write("{0:x2}", remainder[q]);
                Console.WriteLine();
                */

                int j = remainderLen - bi2.dataLength;
                int pos = remainderLen - 1;

                ulong firstDivisorByte = bi2.data[bi2.dataLength-1];
                ulong secondDivisorByte = bi2.data[bi2.dataLength-2];

                int divisorLen = bi2.dataLength + 1;
                uint[] dividendPart = new uint[divisorLen];

                while(j > 0)
                {
                        ulong dividend = ((ulong)remainder[pos] << 32) + (ulong)remainder[pos-1];
                        //Console.WriteLine("dividend = {0}", dividend);

                        ulong q_hat = dividend / firstDivisorByte;
                        ulong r_hat = dividend % firstDivisorByte;

                        //Console.WriteLine("q_hat = {0:X}, r_hat = {1:X}", q_hat, r_hat);

                        bool done = false;
                        while(!done)
                        {
                                done = true;

                                if(q_hat == 0x100000000 ||
                                   (q_hat * secondDivisorByte) > ((r_hat << 32) + remainder[pos-2]))
                                {
                                        q_hat--;
                                        r_hat += firstDivisorByte;

                                        if(r_hat < 0x100000000)
                                                done = false;
                                }
                        }

                        for(int h = 0; h < divisorLen; h++)
                                dividendPart[h] = remainder[pos-h];

                        BigInteger kk = new BigInteger(dividendPart);
                        BigInteger ss = bi2 * (long)q_hat;

                        //Console.WriteLine("ss before = " + ss);
                        while(ss > kk)
                        {
                                q_hat--;
                                ss -= bi2;
                                //Console.WriteLine(ss);
                        }
                        BigInteger yy = kk - ss;

                        //Console.WriteLine("ss = " + ss);
                        //Console.WriteLine("kk = " + kk);
                        //Console.WriteLine("yy = " + yy);

                        for(int h = 0; h < divisorLen; h++)
                                remainder[pos-h] = yy.data[bi2.dataLength-h];

                        /*
                        Console.WriteLine("dividend = ");
                        for(int q = remainderLen - 1; q >= 0; q--)
                                Console.Write("{0:x2}", remainder[q]);
                        Console.WriteLine("\n************ q_hat = {0:X}\n", q_hat);
                        */

                        result[resultPos++] = (uint)q_hat;

                        pos--;
                        j--;
                }

                outQuotient.dataLength = resultPos;
                int y = 0;
                for(int x = outQuotient.dataLength - 1; x >= 0; x--, y++)
                        outQuotient.data[y] = result[x];
                for(; y < maxLength; y++)
                        outQuotient.data[y] = 0;

                while(outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength-1] == 0)
                        outQuotient.dataLength--;

                if(outQuotient.dataLength == 0)
                        outQuotient.dataLength = 1;

                outRemainder.dataLength = shiftRight(remainder, shift);

                for(y = 0; y < outRemainder.dataLength; y++)
                        outRemainder.data[y] = remainder[y];
                for(; y < maxLength; y++)
                        outRemainder.data[y] = 0;
        }


        //***********************************************************************
        // Private function that supports the division of two numbers with
        // a divisor that has only 1 digit.
        //***********************************************************************

        private static void singleByteDivide(BigInteger bi1, BigInteger bi2,
                                             BigInteger outQuotient, BigInteger outRemainder)
        {
                uint[] result = new uint[maxLength];
                int resultPos = 0;

                // copy dividend to reminder
                for(int i = 0; i < maxLength; i++)
                        outRemainder.data[i] = bi1.data[i];
                outRemainder.dataLength = bi1.dataLength;

                while(outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength-1] == 0)
                        outRemainder.dataLength--;

                ulong divisor = (ulong)bi2.data[0];
                int pos = outRemainder.dataLength - 1;
                ulong dividend = (ulong)outRemainder.data[pos];

                //Console.WriteLine("divisor = " + divisor + " dividend = " + dividend);
                //Console.WriteLine("divisor = " + bi2 + "\ndividend = " + bi1);

                if(dividend >= divisor)
                {
                        ulong quotient = dividend / divisor;
                        result[resultPos++] = (uint)quotient;

                        outRemainder.data[pos] = (uint)(dividend % divisor);
                }
                pos--;

                while(pos >= 0)
                {
                        //Console.WriteLine(pos);

                        dividend = ((ulong)outRemainder.data[pos+1] << 32) + (ulong)outRemainder.data[pos];
                        ulong quotient = dividend / divisor;
                        result[resultPos++] = (uint)quotient;

                        outRemainder.data[pos+1] = 0;
                        outRemainder.data[pos--] = (uint)(dividend % divisor);
                        //Console.WriteLine(">>>> " + bi1);
                }

                outQuotient.dataLength = resultPos;
                int j = 0;
                for(int i = outQuotient.dataLength - 1; i >= 0; i--, j++)
                        outQuotient.data[j] = result[i];
                for(; j < maxLength; j++)
                        outQuotient.data[j] = 0;

                while(outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength-1] == 0)
                        outQuotient.dataLength--;

                if(outQuotient.dataLength == 0)
                        outQuotient.dataLength = 1;

                while(outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength-1] == 0)
                        outRemainder.dataLength--;
        }


        //***********************************************************************
        // Overloading of division operator
        //***********************************************************************

        public static BigInteger operator /(BigInteger bi1, BigInteger bi2)
        {
                BigInteger quotient = new BigInteger();
                BigInteger remainder = new BigInteger();

                int lastPos = maxLength-1;
                bool divisorNeg = false, dividendNeg = false;

                if((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
                {
                        bi1 = -bi1;
                        dividendNeg = true;
                }
                if((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
                {
                        bi2 = -bi2;
                        divisorNeg = true;
                }

                if(bi1 < bi2)
                {
                        return quotient;
                }

                else
                {
                        if(bi2.dataLength == 1)
                                singleByteDivide(bi1, bi2, quotient, remainder);
                        else
                                multiByteDivide(bi1, bi2, quotient, remainder);

                        if(dividendNeg != divisorNeg)
                                return -quotient;

                        return quotient;
                }
        }


        //***********************************************************************
        // Overloading of modulus operator
        //***********************************************************************

        public static BigInteger operator %(BigInteger bi1, BigInteger bi2)
        {
                BigInteger quotient = new BigInteger();
                BigInteger remainder = new BigInteger(bi1);

                int lastPos = maxLength-1;
                bool dividendNeg = false;

                if((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
                {
                        bi1 = -bi1;
                        dividendNeg = true;
                }
                if((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
                        bi2 = -bi2;

                if(bi1 < bi2)
                {
                        return remainder;
                }

                else
                {
                        if(bi2.dataLength == 1)
                                singleByteDivide(bi1, bi2, quotient, remainder);
                        else
                                multiByteDivide(bi1, bi2, quotient, remainder);

                        if(dividendNeg)
                                return -remainder;

                        return remainder;
                }
        }


        //***********************************************************************
        // Overloading of bitwise AND operator
        //***********************************************************************

        public static BigInteger operator &(BigInteger bi1, BigInteger bi2)
        {
                BigInteger result = new BigInteger();

                int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

                for(int i = 0; i < len; i++)
                {
                        uint sum = (uint)(bi1.data[i] & bi2.data[i]);
                        result.data[i] = sum;
                }

                result.dataLength = maxLength;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                return result;
        }


        //***********************************************************************
        // Overloading of bitwise OR operator
        //***********************************************************************

        public static BigInteger operator |(BigInteger bi1, BigInteger bi2)
        {
                BigInteger result = new BigInteger();

                int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

                for(int i = 0; i < len; i++)
                {
                        uint sum = (uint)(bi1.data[i] | bi2.data[i]);
                        result.data[i] = sum;
                }

                result.dataLength = maxLength;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                return result;
        }


        //***********************************************************************
        // Overloading of bitwise XOR operator
        //***********************************************************************

        public static BigInteger operator ^(BigInteger bi1, BigInteger bi2)
        {
                BigInteger result = new BigInteger();

                int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

                for(int i = 0; i < len; i++)
                {
                        uint sum = (uint)(bi1.data[i] ^ bi2.data[i]);
                        result.data[i] = sum;
                }

                result.dataLength = maxLength;

                while(result.dataLength > 1 && result.data[result.dataLength-1] == 0)
                        result.dataLength--;

                return result;
        }


        //***********************************************************************
        // Returns max(this, bi)
        //***********************************************************************

        public BigInteger max(BigInteger bi)
        {
                if(this > bi)
                        return (new BigInteger(this));
                else
                        return (new BigInteger(bi));
        }


        //***********************************************************************
        // Returns min(this, bi)
        //***********************************************************************

        public BigInteger min(BigInteger bi)
        {
                if(this < bi)
                        return (new BigInteger(this));
                else
                        return (new BigInteger(bi));

        }


        //***********************************************************************
        // Returns the absolute value
        //***********************************************************************

        public BigInteger abs()
        {
                if((this.data[maxLength - 1] & 0x80000000) != 0)
                        return (-this);
                else
                        return (new BigInteger(this));
        }


        //***********************************************************************
        // Returns a string representing the BigInteger in base 10.
        //***********************************************************************

        public override string ToString()
        {
                return ToString(10);
        }


        //***********************************************************************
        // Returns a string representing the BigInteger in sign-and-magnitude
        // format in the specified radix.
        //
        // Example
        // -------
        // If the value of BigInteger is -255 in base 10, then
        // ToString(16) returns "-FF"
        //
        //***********************************************************************

        public string ToString(int radix)
        {
                if(radix < 2 || radix > 36)
                        throw (new ArgumentException("Radix must be >= 2 and <= 36"));

                string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
                string result = "";

                BigInteger a = this;

                bool negative = false;
                if((a.data[maxLength-1] & 0x80000000) != 0)
                {
                        negative = true;
                        try
                        {
                                a = -a;
                        }
                        catch(Exception) {}
                }

                BigInteger quotient = new BigInteger();
                BigInteger remainder = new BigInteger();
                BigInteger biRadix = new BigInteger(radix);

                if(a.dataLength == 1 && a.data[0] == 0)
                        result = "0";
                else
                {
                        while(a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0))
                        {
                                singleByteDivide(a, biRadix, quotient, remainder);

                                if(remainder.data[0] < 10)
                                        result = remainder.data[0] + result;
                                else
                                        result = charSet[(int)remainder.data[0] - 10] + result;

                                a = quotient;
                        }
                        if(negative)
                                result = "-" + result;
                }

                return result;
        }


        //***********************************************************************
        // Returns a hex string showing the contains of the BigInteger
        //
        // Examples
        // -------
        // 1) If the value of BigInteger is 255 in base 10, then
        //    ToHexString() returns "FF"
        //
        // 2) If the value of BigInteger is -255 in base 10, then
        //    ToHexString() returns ".....FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF01",
        //    which is the 2's complement representation of -255.
        //
        //***********************************************************************

        public string ToHexString()
        {
                string result = data[dataLength - 1].ToString("X");

                for(int i = dataLength - 2; i >= 0; i--)
                {
                        result += data[i].ToString("X8");
                }

                return result;
        }



        //***********************************************************************
        // Modulo Exponentiation
        //***********************************************************************

        public BigInteger modPow(BigInteger exp, BigInteger n)
        {
                if((exp.data[maxLength-1] & 0x80000000) != 0)
                        throw (new ArithmeticException("Positive exponents only."));

                BigInteger resultNum = 1;
	        BigInteger tempNum;
	        bool thisNegative = false;

	        if((this.data[maxLength-1] & 0x80000000) != 0)   // negative this
	        {
	                tempNum = -this % n;
	                thisNegative = true;
	        }
	        else
	                tempNum = this % n;  // ensures (tempNum * tempNum) < b^(2k)

	        if((n.data[maxLength-1] & 0x80000000) != 0)   // negative n
	                n = -n;

                // calculate constant = b^(2k) / m
                BigInteger constant = new BigInteger();

                int i = n.dataLength << 1;
                constant.data[i] = 0x00000001;
                constant.dataLength = i + 1;

                constant = constant / n;
                int totalBits = exp.bitCount();
                int count = 0;

                // perform squaring and multiply exponentiation
                for(int pos = 0; pos < exp.dataLength; pos++)
                {
                        uint mask = 0x01;
                        //Console.WriteLine("pos = " + pos);

                        for(int index = 0; index < 32; index++)
                        {
                                if((exp.data[pos] & mask) != 0)
                                        resultNum = BarrettReduction(resultNum * tempNum, n, constant);

                                mask <<= 1;

                                tempNum = BarrettReduction(tempNum * tempNum, n, constant);


                                if(tempNum.dataLength == 1 && tempNum.data[0] == 1)
                                {
                                        if(thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
                                                return -resultNum;
                                        return resultNum;
                                }
                                count++;
                                if(count == totalBits)
                                        break;
                        }
                }

                if(thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
                        return -resultNum;

	        return resultNum;
        }



        //***********************************************************************
        // Fast calculation of modular reduction using Barrett's reduction.
        // Requires x < b^(2k), where b is the base.  In this case, base is
        // 2^32 (uint).
        //
        // Reference [4]
        //***********************************************************************

        private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
        {
                int k = n.dataLength,
                    kPlusOne = k+1,
                    kMinusOne = k-1;

                BigInteger q1 = new BigInteger();

                // q1 = x / b^(k-1)
                for(int i = kMinusOne, j = 0; i < x.dataLength; i++, j++)
                        q1.data[j] = x.data[i];
                q1.dataLength = x.dataLength - kMinusOne;
                if(q1.dataLength <= 0)
                        q1.dataLength = 1;


                BigInteger q2 = q1 * constant;
                BigInteger q3 = new BigInteger();

                // q3 = q2 / b^(k+1)
                for(int i = kPlusOne, j = 0; i < q2.dataLength; i++, j++)
                        q3.data[j] = q2.data[i];
                q3.dataLength = q2.dataLength - kPlusOne;
                if(q3.dataLength <= 0)
                        q3.dataLength = 1;


                // r1 = x mod b^(k+1)
                // i.e. keep the lowest (k+1) words
                BigInteger r1 = new BigInteger();
                int lengthToCopy = (x.dataLength > kPlusOne) ? kPlusOne : x.dataLength;
                for(int i = 0; i < lengthToCopy; i++)
                        r1.data[i] = x.data[i];
                r1.dataLength = lengthToCopy;


                // r2 = (q3 * n) mod b^(k+1)
                // partial multiplication of q3 and n

                BigInteger r2 = new BigInteger();
                for(int i = 0; i < q3.dataLength; i++)
                {
                        if(q3.data[i] == 0)     continue;

                        ulong mcarry = 0;
                        int t = i;
                        for(int j = 0; j < n.dataLength && t < kPlusOne; j++, t++)
                        {
                                // t = i + j
                                ulong val = ((ulong)q3.data[i] * (ulong)n.data[j]) +
                                             (ulong)r2.data[t] + mcarry;

                                r2.data[t] = (uint)(val & 0xFFFFFFFF);
                                mcarry = (val >> 32);
                        }

                        if(t < kPlusOne)
                                r2.data[t] = (uint)mcarry;
                }
                r2.dataLength = kPlusOne;
                while(r2.dataLength > 1 && r2.data[r2.dataLength-1] == 0)
                        r2.dataLength--;

                r1 -= r2;
                if((r1.data[maxLength-1] & 0x80000000) != 0)        // negative
                {
                        BigInteger val = new BigInteger();
                        val.data[kPlusOne] = 0x00000001;
                        val.dataLength = kPlusOne + 1;
                        r1 += val;
                }

                while(r1 >= n)
                        r1 -= n;

                return r1;
        }


        //***********************************************************************
        // Returns gcd(this, bi)
        //***********************************************************************

        public BigInteger gcd(BigInteger bi)
        {
                BigInteger x;
                BigInteger y;

                if((data[maxLength-1] & 0x80000000) != 0)     // negative
                        x = -this;
                else
                        x = this;

                if((bi.data[maxLength-1] & 0x80000000) != 0)     // negative
                        y = -bi;
                else
                        y = bi;

	        BigInteger g = y;

	        while(x.dataLength > 1 || (x.dataLength == 1 && x.data[0] != 0))
	        {
		        g = x;
		        x = y % x;
		        y = g;
        	}

	        return g;
        }


        //***********************************************************************
        // Populates "this" with the specified amount of random bits
        //***********************************************************************

        public void genRandomBits(int bits, Random rand)
        {
	        int dwords = bits >> 5;
	        int remBits = bits & 0x1F;

	        if(remBits != 0)
		        dwords++;

	        if(dwords > maxLength)
		        throw (new ArithmeticException("Number of required bits > maxLength."));

	        for(int i = 0; i < dwords; i++)
		        data[i] = (uint)(rand.NextDouble() * 0x100000000);

	        for(int i = dwords; i < maxLength; i++)
		        data[i] = 0;

	        if(remBits != 0)
	        {
		        uint mask = (uint)(0x01 << (remBits-1));
		        data[dwords-1] |= mask;

		        mask = (uint)(0xFFFFFFFF >> (32 - remBits));
		        data[dwords-1] &= mask;
	        }
	        else
		        data[dwords-1] |= 0x80000000;

	        dataLength = dwords;

	        if(dataLength == 0)
	                dataLength = 1;
        }


        //***********************************************************************
        // Returns the position of the most significant bit in the BigInteger.
        //
        // Eg.  The result is 0, if the value of BigInteger is 0...0000 0000
        //      The result is 1, if the value of BigInteger is 0...0000 0001
        //      The result is 2, if the value of BigInteger is 0...0000 0010
        //      The result is 2, if the value of BigInteger is 0...0000 0011
        //
        //***********************************************************************

        public int bitCount()
        {
                while(dataLength > 1 && data[dataLength-1] == 0)
                        dataLength--;

               uint value = data[dataLength - 1];
               uint mask = 0x80000000;
               int bits = 32;

               while(bits > 0 && (value & mask) == 0)
               {
                        bits--;
                        mask >>= 1;
               }
               bits += ((dataLength - 1) << 5);

               return bits;
        }




       


        //***********************************************************************
        // Probabilistic prime test based on Rabin-Miller's
        //
        // for any p > 0 with p - 1 = 2^s * t
        //
        // p is probably prime (strong pseudoprime) if for any a < p,
        // 1) a^t mod p = 1 or
        // 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
        //
        // Otherwise, p is composite.
        //
        // Returns
        // -------
        // True if "this" is a strong pseudoprime to randomly chosen
        // bases.  The number of chosen bases is given by the "confidence"
        // parameter.
        //
        // False if "this" is definitely NOT prime.
        //
        //***********************************************************************

        public bool RabinMillerTest(int confidence)
        {
                BigInteger thisVal;
                if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
                        thisVal = -this;
                else
                        thisVal = this;

                if(thisVal.dataLength == 1)
                {
                        // test small numbers
                        if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
                                return false;
                        else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
                                return true;
                }

                if((thisVal.data[0] & 0x1) == 0)     // even numbers
                        return false;


                // calculate values of s and t
                BigInteger p_sub1 = thisVal - (new BigInteger(1));
                int s = 0;

                for(int index = 0; index < p_sub1.dataLength; index++)
                {
                        uint mask = 0x01;

                        for(int i = 0; i < 32; i++)
                        {
                                if((p_sub1.data[index] & mask) != 0)
                                {
                                        index = p_sub1.dataLength;      // to break the outer loop
                                        break;
                                }
                                mask <<= 1;
                                s++;
                        }
                }

                BigInteger t = p_sub1 >> s;

	        int bits = thisVal.bitCount();
	        BigInteger a = new BigInteger();
	        Random rand = new Random();

	        for(int round = 0; round < confidence; round++)
	        {
		        bool done = false;

		        while(!done)		// generate a < n
		        {
			        int testBits = 0;

			        // make sure "a" has at least 2 bits
			        while(testBits < 2)
				        testBits = (int)(rand.NextDouble() * bits);

			        a.genRandomBits(testBits, rand);

			        int byteLen = a.dataLength;

                                // make sure "a" is not 0
			        if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
				        done = true;
		        }

                        // check whether a factor exists (fix for version 1.03)
		        BigInteger gcdTest = a.gcd(thisVal);
                        if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
                                return false;

                        BigInteger b = a.modPow(t, thisVal);

                        /*
                        Console.WriteLine("a = " + a.ToString(10));
                        Console.WriteLine("b = " + b.ToString(10));
                        Console.WriteLine("t = " + t.ToString(10));
                        Console.WriteLine("s = " + s);
                        */

                        bool result = false;

                        if(b.dataLength == 1 && b.data[0] == 1)         // a^t mod p = 1
                                result = true;

                        for(int j = 0; result == false && j < s; j++)
                        {
                                if(b == p_sub1)         // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
                                {
                                        result = true;
                                        break;
                                }

                                b = (b * b) % thisVal;
                        }

                        if(result == false)
                                return false;
                }
	        return true;
        }


        //***********************************************************************
        // Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
        //
        // p is probably prime if for any a < p (a is not multiple of p),
        // a^((p-1)/2) mod p = J(a, p)
        //
        // where J is the Jacobi symbol.
        //
        // Otherwise, p is composite.
        //
        // Returns
        // -------
        // True if "this" is a Euler pseudoprime to randomly chosen
        // bases.  The number of chosen bases is given by the "confidence"
        // parameter.
        //
        // False if "this" is definitely NOT prime.
        //
        //***********************************************************************

        public bool SolovayStrassenTest(int confidence)
        {
                BigInteger thisVal;
                if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
                        thisVal = -this;
                else
                        thisVal = this;

                if(thisVal.dataLength == 1)
                {
                        // test small numbers
                        if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
                                return false;
                        else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
                                return true;
                }

                if((thisVal.data[0] & 0x1) == 0)     // even numbers
                        return false;


	        int bits = thisVal.bitCount();
	        BigInteger a = new BigInteger();
	        BigInteger p_sub1 = thisVal - 1;
	        BigInteger p_sub1_shift = p_sub1 >> 1;

	        Random rand = new Random();

	        for(int round = 0; round < confidence; round++)
	        {
		        bool done = false;

		        while(!done)		// generate a < n
		        {
			        int testBits = 0;

			        // make sure "a" has at least 2 bits
			        while(testBits < 2)
				        testBits = (int)(rand.NextDouble() * bits);

			        a.genRandomBits(testBits, rand);

			        int byteLen = a.dataLength;

                                // make sure "a" is not 0
			        if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
				        done = true;
		        }

                        // check whether a factor exists (fix for version 1.03)
		        BigInteger gcdTest = a.gcd(thisVal);
                        if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
                                return false;

		        // calculate a^((p-1)/2) mod p

		        BigInteger expResult = a.modPow(p_sub1_shift, thisVal);
		        if(expResult == p_sub1)
		                expResult = -1;

                        // calculate Jacobi symbol
                        BigInteger jacob = Jacobi(a, thisVal);

                        //Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
                        //Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10));

                        // if they are different then it is not prime
                        if(expResult != jacob)
			        return false;
	        }

	        return true;
        }


        //***********************************************************************
        // Implementation of the Lucas Strong Pseudo Prime test.
        //
        // Let n be an odd number with gcd(n,D) = 1, and n - J(D, n) = 2^s * d
        // with d odd and s >= 0.
        //
        // If Ud mod n = 0 or V2^r*d mod n = 0 for some 0 <= r < s, then n
        // is a strong Lucas pseudoprime with parameters (P, Q).  We select
        // P and Q based on Selfridge.
        //
        // Returns True if number is a strong Lucus pseudo prime.
        // Otherwise, returns False indicating that number is composite.
        //***********************************************************************

        public bool LucasStrongTest()
        {
                BigInteger thisVal;
                if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
                        thisVal = -this;
                else
                        thisVal = this;

                if(thisVal.dataLength == 1)
                {
                        // test small numbers
                        if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
                                return false;
                        else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
                                return true;
                }

                if((thisVal.data[0] & 0x1) == 0)     // even numbers
                        return false;

                return LucasStrongTestHelper(thisVal);
        }


        private bool LucasStrongTestHelper(BigInteger thisVal)
        {
                // Do the test (selects D based on Selfridge)
                // Let D be the first element of the sequence
                // 5, -7, 9, -11, 13, ... for which J(D,n) = -1
                // Let P = 1, Q = (1-D) / 4

                long D = 5, sign = -1, dCount = 0;
                bool done = false;

                while(!done)
                {
                        int Jresult = BigInteger.Jacobi(D, thisVal);

                        if(Jresult == -1)
                                done = true;    // J(D, this) = 1
                        else
                        {
                                if(Jresult == 0 && Math.Abs(D) < thisVal)       // divisor found
                                        return false;

                                if(dCount == 20)
                                {
                                        // check for square
                                        BigInteger root = thisVal.sqrt();
                                        if(root * root == thisVal)
                                                return false;
                                }

                                //Console.WriteLine(D);
                                D = (Math.Abs(D) + 2) * sign;
                                sign = -sign;
                        }
                        dCount++;
                }

                long Q = (1 - D) >> 2;

                /*
                Console.WriteLine("D = " + D);
                Console.WriteLine("Q = " + Q);
                Console.WriteLine("(n,D) = " + thisVal.gcd(D));
                Console.WriteLine("(n,Q) = " + thisVal.gcd(Q));
                Console.WriteLine("J(D|n) = " + BigInteger.Jacobi(D, thisVal));
                */

                BigInteger p_add1 = thisVal + 1;
                int s = 0;

                for(int index = 0; index < p_add1.dataLength; index++)
                {
                        uint mask = 0x01;

                        for(int i = 0; i < 32; i++)
                        {
                                if((p_add1.data[index] & mask) != 0)
                                {
                                        index = p_add1.dataLength;      // to break the outer loop
                                        break;
                                }
                                mask <<= 1;
                                s++;
                        }
                }

                BigInteger t = p_add1 >> s;

                // calculate constant = b^(2k) / m
                // for Barrett Reduction
                BigInteger constant = new BigInteger();

                int nLen = thisVal.dataLength << 1;
                constant.data[nLen] = 0x00000001;
                constant.dataLength = nLen + 1;

                constant = constant / thisVal;

                BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0);
                bool isPrime = false;

                if((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) ||
                   (lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
                {
                        // u(t) = 0 or V(t) = 0
                        isPrime = true;
                }

                for(int i = 1; i < s; i++)
                {
                        if(!isPrime)
                        {
                                // doubling of index
                                lucas[1] = thisVal.BarrettReduction(lucas[1] * lucas[1], thisVal, constant);
                                lucas[1] = (lucas[1] - (lucas[2] << 1)) % thisVal;

                                //lucas[1] = ((lucas[1] * lucas[1]) - (lucas[2] << 1)) % thisVal;

                                if((lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
                                        isPrime = true;
                        }

                        lucas[2] = thisVal.BarrettReduction(lucas[2] * lucas[2], thisVal, constant);     //Q^k
                }


                if(isPrime)     // additional checks for composite numbers
                {
                        // If n is prime and gcd(n, Q) == 1, then
                        // Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n

                        BigInteger g = thisVal.gcd(Q);
                        if(g.dataLength == 1 && g.data[0] == 1)         // gcd(this, Q) == 1
                        {
                                if((lucas[2].data[maxLength-1] & 0x80000000) != 0)
                                        lucas[2] += thisVal;

                                BigInteger temp = (Q * BigInteger.Jacobi(Q, thisVal)) % thisVal;
                                if((temp.data[maxLength-1] & 0x80000000) != 0)
                                        temp += thisVal;

                                if(lucas[2] != temp)
                                        isPrime = false;
                        }
                }

                return isPrime;
        }


        //***********************************************************************
        // Determines whether a number is probably prime, using the Rabin-Miller's
        // test.  Before applying the test, the number is tested for divisibility
        // by primes < 2000
        //
        // Returns true if number is probably prime.
        //***********************************************************************

        public bool isProbablePrime(int confidence)
        {
                BigInteger thisVal;
                if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
                        thisVal = -this;
                else
                        thisVal = this;


                // test for divisibility by primes < 2000
                for(int p = 0; p < primesBelow2000.Length; p++)
                {
                        BigInteger divisor = primesBelow2000[p];

                        if(divisor >= thisVal)
                                break;

                        BigInteger resultNum = thisVal % divisor;
                        if(resultNum.IntValue() == 0)
                        {
                                /*
				Console.WriteLine("Not prime!  Divisible by {0}\n",
                                                  primesBelow2000[p]);
                                */
                                return false;
                        }
                }

                if(thisVal.RabinMillerTest(confidence))
                        return true;
                else
                {
                        //Console.WriteLine("Not prime!  Failed primality test\n");
                        return false;
                }
        }


        //***********************************************************************
        // Determines whether this BigInteger is probably prime using a
        // combination of base 2 strong pseudoprime test and Lucas strong
        // pseudoprime test.
        //
        // The sequence of the primality test is as follows,
        //
        // 1) Trial divisions are carried out using prime numbers below 2000.
        //    if any of the primes divides this BigInteger, then it is not prime.
        //
        // 2) Perform base 2 strong pseudoprime test.  If this BigInteger is a
        //    base 2 strong pseudoprime, proceed on to the next step.
        //
        // 3) Perform strong Lucas pseudoprime test.
        //
        // Returns True if this BigInteger is both a base 2 strong pseudoprime
        // and a strong Lucas pseudoprime.
        //
        // For a detailed discussion of this primality test, see [6].
        //
        //***********************************************************************

        public bool isProbablePrime()
        {
                BigInteger thisVal;
                if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
                        thisVal = -this;
                else
                        thisVal = this;

                if(thisVal.dataLength == 1)
                {
                        // test small numbers
                        if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
                                return false;
                        else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
                                return true;
                }

                if((thisVal.data[0] & 0x1) == 0)     // even numbers
                        return false;


                // test for divisibility by primes < 2000
                for(int p = 0; p < primesBelow2000.Length; p++)
                {
                        BigInteger divisor = primesBelow2000[p];

                        if(divisor >= thisVal)
                                break;

                        BigInteger resultNum = thisVal % divisor;
                        if(resultNum.IntValue() == 0)
                        {
				//Console.WriteLine("Not prime!  Divisible by {0}\n",
                                //                  primesBelow2000[p]);

                                return false;
                        }
                }

                // Perform BASE 2 Rabin-Miller Test

                // calculate values of s and t
                BigInteger p_sub1 = thisVal - (new BigInteger(1));
                int s = 0;

                for(int index = 0; index < p_sub1.dataLength; index++)
                {
                        uint mask = 0x01;

                        for(int i = 0; i < 32; i++)
                        {
                                if((p_sub1.data[index] & mask) != 0)
                                {
                                        index = p_sub1.dataLength;      // to break the outer loop
                                        break;
                                }
                                mask <<= 1;
                                s++;
                        }
                }

                BigInteger t = p_sub1 >> s;

	        int bits = thisVal.bitCount();
	        BigInteger a = 2;

                // b = a^t mod p
                BigInteger b = a.modPow(t, thisVal);
                bool result = false;

                if(b.dataLength == 1 && b.data[0] == 1)         // a^t mod p = 1
                        result = true;

                for(int j = 0; result == false && j < s; j++)
                {
                        if(b == p_sub1)         // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
                        {
                                result = true;
                                break;
                        }

                        b = (b * b) % thisVal;
                }

                // if number is strong pseudoprime to base 2, then do a strong lucas test
                if(result)
	                result = LucasStrongTestHelper(thisVal);

                return result;
        }



        //***********************************************************************
        // Returns the lowest 4 bytes of the BigInteger as an int.
        //***********************************************************************

        public int IntValue()
        {
                return (int)data[0];
        }


        //***********************************************************************
        // Returns the lowest 8 bytes of the BigInteger as a long.
        //***********************************************************************

        public long LongValue()
        {
                long val = 0;

                val = (long)data[0];
                try
                {       // exception if maxLength = 1
                        val |= (long)data[1] << 32;
                }
                catch(Exception)
                {
                        if((data[0] & 0x80000000) != 0) // negative
                                val = (int)data[0];
                }

                return val;
        }


        //***********************************************************************
        // Computes the Jacobi Symbol for a and b.
        // Algorithm adapted from [3] and [4] with some optimizations
        //***********************************************************************

        public static int Jacobi(BigInteger a, BigInteger b)
        {
                // Jacobi defined only for odd integers
                if((b.data[0] & 0x1) == 0)
                        throw (new ArgumentException("Jacobi defined only for odd integers."));

                if(a >= b)      a %= b;
                if(a.dataLength == 1 && a.data[0] == 0)      return 0;  // a == 0
                if(a.dataLength == 1 && a.data[0] == 1)      return 1;  // a == 1

                if(a < 0)
                {
                        if( (((b-1).data[0]) & 0x2) == 0)       //if( (((b-1) >> 1).data[0] & 0x1) == 0)
                                return Jacobi(-a, b);
                        else
                                return -Jacobi(-a, b);
                }

                int e = 0;
                for(int index = 0; index < a.dataLength; index++)
                {
                        uint mask = 0x01;

                        for(int i = 0; i < 32; i++)
                        {
                                if((a.data[index] & mask) != 0)
                                {
                                        index = a.dataLength;      // to break the outer loop
                                        break;
                                }
                                mask <<= 1;
                                e++;
                        }
                }

                BigInteger a1 = a >> e;

                int s = 1;
                if((e & 0x1) != 0 && ((b.data[0] & 0x7) == 3 || (b.data[0] & 0x7) == 5))
                        s = -1;

                if((b.data[0] & 0x3) == 3 && (a1.data[0] & 0x3) == 3)
                        s = -s;

                if(a1.dataLength == 1 && a1.data[0] == 1)
                        return s;
                else
                        return (s * Jacobi(b % a1, a1));
        }



        //***********************************************************************
        // Generates a positive BigInteger that is probably prime.
        //***********************************************************************

        public static BigInteger genPseudoPrime(int bits, int confidence, Random rand)
        {
	        BigInteger result = new BigInteger();
	        bool done = false;

	        while(!done)
	        {
		        result.genRandomBits(bits, rand);
		        result.data[0] |= 0x01;		// make it odd

		        // prime test
		        done = result.isProbablePrime(confidence);
	        }
	        return result;
        }


        //***********************************************************************
        // Generates a random number with the specified number of bits such
        // that gcd(number, this) = 1
        //***********************************************************************

        public BigInteger genCoPrime(int bits, Random rand)
        {
	        bool done = false;
	        BigInteger result = new BigInteger();

	        while(!done)
	        {
	                result.genRandomBits(bits, rand);
	                //Console.WriteLine(result.ToString(16));

		        // gcd test
		        BigInteger g = result.gcd(this);
			if(g.dataLength == 1 && g.data[0] == 1)
                                done = true;
	        }

	        return result;
        }


        //***********************************************************************
        // Returns the modulo inverse of this.  Throws ArithmeticException if
        // the inverse does not exist.  (i.e. gcd(this, modulus) != 1)
        //***********************************************************************

        public BigInteger modInverse(BigInteger modulus)
        {
                BigInteger[] p = { 0, 1 };
                BigInteger[] q = new BigInteger[2];    // quotients
                BigInteger[] r = { 0, 0 };             // remainders

                int step = 0;

                BigInteger a = modulus;
                BigInteger b = this;

                while(b.dataLength > 1 || (b.dataLength == 1 && b.data[0] != 0))
                {
                        BigInteger quotient = new BigInteger();
                        BigInteger remainder = new BigInteger();

                        if(step > 1)
                        {
                                BigInteger pval = (p[0] - (p[1] * q[0])) % modulus;
                                p[0] = p[1];
                                p[1] = pval;
                        }

                        if(b.dataLength == 1)
                                singleByteDivide(a, b, quotient, remainder);
                        else
                                multiByteDivide(a, b, quotient, remainder);

                        /*
                        Console.WriteLine(quotient.dataLength);
                        Console.WriteLine("{0} = {1}({2}) + {3}  p = {4}", a.ToString(10),
                                          b.ToString(10), quotient.ToString(10), remainder.ToString(10),
                                          p[1].ToString(10));
                        */

                        q[0] = q[1];
                        r[0] = r[1];
                        q[1] = quotient; r[1] = remainder;

                        a = b;
                        b = remainder;

                        step++;
                }

                if(r[0].dataLength > 1 || (r[0].dataLength == 1 && r[0].data[0] != 1))
                        throw (new ArithmeticException("No inverse!"));

                BigInteger result = ((p[0] - (p[1] * q[0])) % modulus);

                if((result.data[maxLength - 1] & 0x80000000) != 0)
                        result += modulus;  // get the least positive modulus

                return result;
        }


        //***********************************************************************
        // Returns the value of the BigInteger as a byte array.  The lowest
        // index contains the MSB.
        //***********************************************************************

        public byte[] getBytes()
        {
                int numBits = bitCount();

                int numBytes = numBits >> 3;
                if((numBits & 0x7) != 0)
                        numBytes++;

                byte[] result = new byte[numBytes];

                //Console.WriteLine(result.Length);

                int pos = 0;
                uint tempVal, val = data[dataLength - 1];

                if((tempVal = (val >> 24 & 0xFF)) != 0)
                        result[pos++] = (byte)tempVal;
                if((tempVal = (val >> 16 & 0xFF)) != 0)
                        result[pos++] = (byte)tempVal;
                if((tempVal = (val >> 8 & 0xFF)) != 0)
                        result[pos++] = (byte)tempVal;
                if((tempVal = (val & 0xFF)) != 0)
                        result[pos++] = (byte)tempVal;

                for(int i = dataLength - 2; i >= 0; i--, pos += 4)
                {
                        val = data[i];
                        result[pos+3] = (byte)(val & 0xFF);
                        val >>= 8;
                        result[pos+2] = (byte)(val & 0xFF);
                        val >>= 8;
                        result[pos+1] = (byte)(val & 0xFF);
                        val >>= 8;
                        result[pos] = (byte)(val & 0xFF);
                }

                return result;
        }


        //***********************************************************************
        // Sets the value of the specified bit to 1
        // The Least Significant Bit position is 0.
        //***********************************************************************

        public void setBit(uint bitNum)
        {
                uint bytePos = bitNum >> 5;             // divide by 32
                byte bitPos = (byte)(bitNum & 0x1F);    // get the lowest 5 bits

                uint mask = (uint)1 << bitPos;
                this.data[bytePos] |= mask;

                if(bytePos >= this.dataLength)
                        this.dataLength = (int)bytePos + 1;
        }


        //***********************************************************************
        // Sets the value of the specified bit to 0
        // The Least Significant Bit position is 0.
        //***********************************************************************

        public void unsetBit(uint bitNum)
        {
                uint bytePos = bitNum >> 5;

                if(bytePos < this.dataLength)
                {
                        byte bitPos = (byte)(bitNum & 0x1F);

                        uint mask = (uint)1 << bitPos;
                        uint mask2 = 0xFFFFFFFF ^ mask;

                        this.data[bytePos] &= mask2;

                        if(this.dataLength > 1 && this.data[this.dataLength - 1] == 0)
                                this.dataLength--;
                }
        }


        //***********************************************************************
        // Returns a value that is equivalent to the integer square root
        // of the BigInteger.
        //
        // The integer square root of "this" is defined as the largest integer n
        // such that (n * n) <= this
        //
        //***********************************************************************

        public BigInteger sqrt()
        {
                uint numBits = (uint)this.bitCount();

                if((numBits & 0x1) != 0)        // odd number of bits
                        numBits = (numBits >> 1) + 1;
                else
                        numBits = (numBits >> 1);

                uint bytePos = numBits >> 5;
                byte bitPos = (byte)(numBits & 0x1F);

                uint mask;

                BigInteger result = new BigInteger();
                if(bitPos == 0)
                        mask = 0x80000000;
                else
                {
                        mask = (uint)1 << bitPos;
                        bytePos++;
                }
                result.dataLength = (int)bytePos;

                for(int i = (int)bytePos - 1; i >= 0; i--)
                {
                        while(mask != 0)
                        {
                                // guess
                                result.data[i] ^= mask;

                                // undo the guess if its square is larger than this
                                if((result * result) > this)
                                        result.data[i] ^= mask;

                                mask >>= 1;
                        }
                        mask = 0x80000000;
                }
                return result;
        }


        //***********************************************************************
        // Returns the k_th number in the Lucas Sequence reduced modulo n.
        //
        // Uses index doubling to speed up the process.  For example, to calculate V(k),
        // we maintain two numbers in the sequence V(n) and V(n+1).
        //
        // To obtain V(2n), we use the identity
        //      V(2n) = (V(n) * V(n)) - (2 * Q^n)
        // To obtain V(2n+1), we first write it as
        //      V(2n+1) = V((n+1) + n)
        // and use the identity
        //      V(m+n) = V(m) * V(n) - Q * V(m-n)
        // Hence,
        //      V((n+1) + n) = V(n+1) * V(n) - Q^n * V((n+1) - n)
        //                   = V(n+1) * V(n) - Q^n * V(1)
        //                   = V(n+1) * V(n) - Q^n * P
        //
        // We use k in its binary expansion and perform index doubling for each
        // bit position.  For each bit position that is set, we perform an
        // index doubling followed by an index addition.  This means that for V(n),
        // we need to update it to V(2n+1).  For V(n+1), we need to update it to
        // V((2n+1)+1) = V(2*(n+1))
        //
        // This function returns
        // [0] = U(k)
        // [1] = V(k)
        // [2] = Q^n
        //
        // Where U(0) = 0 % n, U(1) = 1 % n
        //       V(0) = 2 % n, V(1) = P % n
        //***********************************************************************

        public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q,
                                                 BigInteger k, BigInteger n)
        {
                if(k.dataLength == 1 && k.data[0] == 0)
                {
                        BigInteger[] result = new BigInteger[3];

                        result[0] = 0; result[1] = 2 % n; result[2] = 1 % n;
                        return result;
                }

                // calculate constant = b^(2k) / m
                // for Barrett Reduction
                BigInteger constant = new BigInteger();

                int nLen = n.dataLength << 1;
                constant.data[nLen] = 0x00000001;
                constant.dataLength = nLen + 1;

                constant = constant / n;

                // calculate values of s and t
                int s = 0;

                for(int index = 0; index < k.dataLength; index++)
                {
                        uint mask = 0x01;

                        for(int i = 0; i < 32; i++)
                        {
                                if((k.data[index] & mask) != 0)
                                {
                                        index = k.dataLength;      // to break the outer loop
                                        break;
                                }
                                mask <<= 1;
                                s++;
                        }
                }

                BigInteger t = k >> s;

                //Console.WriteLine("s = " + s + " t = " + t);
                return LucasSequenceHelper(P, Q, t, n, constant, s);
        }


        //***********************************************************************
        // Performs the calculation of the kth term in the Lucas Sequence.
        // For details of the algorithm, see reference [9].
        //
        // k must be odd.  i.e LSB == 1
        //***********************************************************************

        private static BigInteger[] LucasSequenceHelper(BigInteger P, BigInteger Q,
                                                        BigInteger k, BigInteger n,
                                                        BigInteger constant, int s)
        {
                BigInteger[] result = new BigInteger[3];

                if((k.data[0] & 0x00000001) == 0)
                        throw (new ArgumentException("Argument k must be odd."));

                int numbits = k.bitCount();
                uint mask = (uint)0x1 << ((numbits & 0x1F) - 1);

                // v = v0, v1 = v1, u1 = u1, Q_k = Q^0

                BigInteger v = 2 % n, Q_k = 1 % n,
                           v1 = P % n, u1 = Q_k;
                bool flag = true;

                for(int i = k.dataLength - 1; i >= 0 ; i--)     // iterate on the binary expansion of k
                {
                        //Console.WriteLine("round");
                        while(mask != 0)
                        {
                                if(i == 0 && mask == 0x00000001)        // last bit
                                        break;

                                if((k.data[i] & mask) != 0)             // bit is set
                                {
                                        // index doubling with addition

                                        u1 = (u1 * v1) % n;

                                        v = ((v * v1) - (P * Q_k)) % n;
                                        v1 = n.BarrettReduction(v1 * v1, n, constant);
                                        v1 = (v1 - ((Q_k * Q) << 1)) % n;

                                        if(flag)
                                                flag = false;
                                        else
                                                Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);

                                        Q_k = (Q_k * Q) % n;
                                }
                                else
                                {
                                        // index doubling
                                        u1 = ((u1 * v) - Q_k) % n;

                                        v1 = ((v * v1) - (P * Q_k)) % n;
                                        v = n.BarrettReduction(v * v, n, constant);
                                        v = (v - (Q_k << 1)) % n;

                                        if(flag)
                                        {
                                                Q_k = Q % n;
                                                flag = false;
                                        }
                                        else
                                                Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
                               }

                               mask >>= 1;
                        }
                        mask = 0x80000000;
                }

                // at this point u1 = u(n+1) and v = v(n)
                // since the last bit always 1, we need to transform u1 to u(2n+1) and v to v(2n+1)

                u1 = ((u1 * v) - Q_k) % n;
                v = ((v * v1) - (P * Q_k)) % n;
                if(flag)
                        flag = false;
                else
                        Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);

                Q_k = (Q_k * Q) % n;


                for(int i = 0; i < s; i++)
                {
                        // index doubling
                        u1 = (u1 * v) % n;
                        v = ((v * v) - (Q_k << 1)) % n;

                        if(flag)
                        {
                                Q_k = Q % n;
                                flag = false;
                        }
                        else
                                Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
                }

                result[0] = u1;
                result[1] = v;
                result[2] = Q_k;

                return result;
        }


        //***********************************************************************
        // Tests the correct implementation of the /, %, * and + operators
        //***********************************************************************

        public static void MulDivTest(int rounds)
        {
                Random rand = new Random();
	        byte[] val = new byte[64];
	        byte[] val2 = new byte[64];

	        for(int count = 0; count < rounds; count++)
	        {
	                // generate 2 numbers of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 65);

		        int t2 = 0;
		        while(t2 == 0)
			        t2 = (int)(rand.NextDouble() * 65);

		        bool done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t1)
					        val[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val[i] = 0;

				        if(val[i] != 0)
					        done = true;
			        }
		        }

		        done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t2)
					        val2[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val2[i] = 0;

				        if(val2[i] != 0)
					        done = true;
			        }
		        }

		        while(val[0] == 0)
		                val[0] = (byte)(rand.NextDouble() * 256);
		        while(val2[0] == 0)
		                val2[0] = (byte)(rand.NextDouble() * 256);

                        Console.WriteLine(count);
		        BigInteger bn1 = new BigInteger(val, t1);
		        BigInteger bn2 = new BigInteger(val2, t2);


                        // Determine the quotient and remainder by dividing
                        // the first number by the second.

		        BigInteger bn3 = bn1 / bn2;
		        BigInteger bn4 = bn1 % bn2;

		        // Recalculate the number
		        BigInteger bn5 = (bn3 * bn2) + bn4;

                        // Make sure they're the same
        		if(bn5 != bn1)
	        	{
		        	Console.WriteLine("Error at " + count);
                                Console.WriteLine(bn1 + "\n");
			        Console.WriteLine(bn2 + "\n");
			        Console.WriteLine(bn3 + "\n");
                                Console.WriteLine(bn4 + "\n");
			        Console.WriteLine(bn5 + "\n");
			        return;
		        }
	        }
        }


        //***********************************************************************
        // Tests the correct implementation of the modulo exponential function
        // using RSA encryption and decryption (using pre-computed encryption and
        // decryption keys).
        //***********************************************************************

        public static void RSATest(int rounds)
        {
                Random rand = new Random(1);
	        byte[] val = new byte[64];

	        // private and public key
                BigInteger bi_e = new BigInteger("a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7", 16);
                BigInteger bi_d = new BigInteger("4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7", 16);
                BigInteger bi_n = new BigInteger("e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f", 16);

                Console.WriteLine("e =\n" + bi_e.ToString(10));
                Console.WriteLine("\nd =\n" + bi_d.ToString(10));
                Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");

	        for(int count = 0; count < rounds; count++)
	        {
	                // generate data of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 65);

		        bool done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t1)
					        val[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val[i] = 0;

				        if(val[i] != 0)
					        done = true;
			        }
		        }

		        while(val[0] == 0)
		                val[0] = (byte)(rand.NextDouble() * 256);

                        Console.Write("Round = " + count);

                        // encrypt and decrypt data
		        BigInteger bi_data = new BigInteger(val, t1);
                        BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
                        BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);

                        // compare
                        if(bi_decrypted != bi_data)
	        	{
		        	Console.WriteLine("\nError at round " + count);
                                Console.WriteLine(bi_data + "\n");
			        return;
		        }
		        Console.WriteLine(" <PASSED>.");
	        }

        }


        //***********************************************************************
        // Tests the correct implementation of the modulo exponential and
        // inverse modulo functions using RSA encryption and decryption.  The two
        // pseudoprimes p and q are fixed, but the two RSA keys are generated
        // for each round of testing.
        //***********************************************************************

        public static void RSATest2(int rounds)
        {
                Random rand = new Random();
	        byte[] val = new byte[64];

                byte[] pseudoPrime1 = {
                        (byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
                        (byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
                        (byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
                        (byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
                        (byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
                        (byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
                        (byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
                        (byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
                        (byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
                        (byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
                        (byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
                };

                byte[] pseudoPrime2 = {
                        (byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
                        (byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
                        (byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
                        (byte)0x81, (byte)0xCD, (byte)0xDE, (byte)0x42, (byte)0xDC, (byte)0x93,
                        (byte)0xC2, (byte)0x7A, (byte)0x62, (byte)0xAC, (byte)0x6C, (byte)0xAF,
                        (byte)0xDE, (byte)0x74, (byte)0xE3, (byte)0xCB, (byte)0x60, (byte)0x20,
                        (byte)0x38, (byte)0x9C, (byte)0x21, (byte)0xC3, (byte)0xDC, (byte)0xC8,
                        (byte)0xA2, (byte)0x4D, (byte)0xC6, (byte)0x2A, (byte)0x35, (byte)0x7F,
                        (byte)0xF3, (byte)0xA9, (byte)0xE8, (byte)0x1D, (byte)0x7B, (byte)0x2C,
                        (byte)0x78, (byte)0xFA, (byte)0xB8, (byte)0x02, (byte)0x55, (byte)0x80,
                        (byte)0x9B, (byte)0xC2, (byte)0xA5, (byte)0xCB,
                };


                BigInteger bi_p = new BigInteger(pseudoPrime1);
                BigInteger bi_q = new BigInteger(pseudoPrime2);
                BigInteger bi_pq = (bi_p-1)*(bi_q-1);
                BigInteger bi_n = bi_p * bi_q;

	        for(int count = 0; count < rounds; count++)
	        {
	                // generate private and public key
                        BigInteger bi_e = bi_pq.genCoPrime(512, rand);
                        BigInteger bi_d = bi_e.modInverse(bi_pq);

                        Console.WriteLine("\ne =\n" + bi_e.ToString(10));
                        Console.WriteLine("\nd =\n" + bi_d.ToString(10));
                        Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");

	                // generate data of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 65);

		        bool done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t1)
					        val[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val[i] = 0;

				        if(val[i] != 0)
					        done = true;
			        }
		        }

		        while(val[0] == 0)
		                val[0] = (byte)(rand.NextDouble() * 256);

                        Console.Write("Round = " + count);

                        // encrypt and decrypt data
		        BigInteger bi_data = new BigInteger(val, t1);
                        BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
                        BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);

                        // compare
                        if(bi_decrypted != bi_data)
	        	{
		        	Console.WriteLine("\nError at round " + count);
                                Console.WriteLine(bi_data + "\n");
			        return;
		        }
		        Console.WriteLine(" <PASSED>.");
	        }

        }


        //***********************************************************************
        // Tests the correct implementation of sqrt() method.
        //***********************************************************************

        public static void SqrtTest(int rounds)
        {
                Random rand = new Random();
	        for(int count = 0; count < rounds; count++)
	        {
	                // generate data of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 1024);

                        Console.Write("Round = " + count);

                        BigInteger a = new BigInteger();
                        a.genRandomBits(t1, rand);

                        BigInteger b = a.sqrt();
                        BigInteger c = (b+1)*(b+1);

                        // check that b is the largest integer such that b*b <= a
                        if(c <= a)
	        	{
		        	Console.WriteLine("\nError at round " + count);
                                Console.WriteLine(a + "\n");
			        return;
		        }
		        Console.WriteLine(" <PASSED>.");
	        }
        }
        public bool FermatLittleTest(int confidence)
        {
            BigInteger thisVal;
            if ((this.data[maxLength - 1] & 0x80000000) != 0)        
                thisVal = -this;
            else
                thisVal = this;

            if (thisVal.dataLength == 1)
            {
                // kiểm tra số nhỏ
                if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
                    return false;
                else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
                    return true;
            }

            if ((thisVal.data[0] & 0x1) == 0)     // nêu n là số chẵn
                return false;

            int bits = thisVal.bitCount();
            BigInteger a = new BigInteger();
            BigInteger p_sub1 = thisVal - (new BigInteger(1));
            Random rand = new Random();

            for (int round = 0; round < confidence; round++)
            {
                bool done = false;

                while (!done)		// phát sinh a < n
                {
                    int testBits = 0;
                    
                    while (testBits < 2)
                        testBits = (int)(rand.NextDouble() * bits);

                    a.genRandomBits(testBits, rand);

                    int byteLen = a.dataLength;

                    
                    if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
                        done = true;
                }

               
                //BigInteger gcdTest = a.gcd(thisVal);
                //if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
                //    return false;

                // tính a^(n-1) mod n
                BigInteger expResult = a.modPow(p_sub1, thisVal);

                int resultLen = expResult.dataLength;

                // Không phải là số nguyên tố khi a^(n-1) mod n != 1

                if (resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1))
                {

                    return false;
                }
            }

            return true;
        }






        internal string ShowDetail(int confidence)
        {
            BigInteger thisVal;
            string detail = "";
            if ((this.data[maxLength - 1] & 0x80000000) != 0)
                thisVal = -this;
            else
                thisVal = this;

            if (thisVal.dataLength == 1)
            {
                // kiểm tra số nhỏ
                if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
                {
                    detail += "Return FALSE because (" + thisVal + "==0|| " + thisVal + "==1) is True";
                    return detail;
                }
                else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
                {
                    detail += "Return TRUE because (" + thisVal + "==2|| " + thisVal + "==3) is True";
                    return detail;
                }
            }

            if ((thisVal.data[0] & 0x1) == 0)     // nếu n là số chẵn
            {
                detail += "Return FALSE because "+thisVal+" mod 2=0";
                return detail;
            }

            int bits = thisVal.bitCount();
            BigInteger a = new BigInteger();
            BigInteger p_sub1 = thisVal - (new BigInteger(1));
            Random rand = new Random();

            for (int round = 0; round < confidence; round++)
            {
                bool done = false;

                while (!done)		// phát sinh a < n
                {
                    int testBits = 0;

                    while (testBits < 2)
                        testBits = (int)(rand.NextDouble() * bits);

                    a.genRandomBits(testBits, rand);

                    int byteLen = a.dataLength;


                    if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
                        done = true;
                }

                detail += "Step " + (round + 1) + ": random a=" + a + "\r\n";
                //BigInteger gcdTest = a.gcd(thisVal);
                //if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
                //{
                //    detail += "Return FALSE because N and a have GCD";
                //    return detail;
                //    //return false;
                //}

                // tính a^(n-1) mod n
                BigInteger expResult = a.modPow(p_sub1, thisVal);

                int resultLen = expResult.dataLength;

                // Không phải là số nguyên tố khi a^(n-1) mod n != 1
                detail +=  a + "^(" + thisVal + "-1) mod " + thisVal + "=" + expResult + "\r\n";
                if (resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1))
                {
                    detail += "Return FALSE because (a^(n-1) mod n != 1) is true ";
                    return detail;
                    //return false;
                }
            }
            detail += "Return TRUE because test "+confidence+" times is False";
            return detail;
            //return true;
        }
}
